A Multifaceted, Multilevel Exploration of the Physics of Sports

Sports Physics. Edited by Christophe Clanet, École Polytechnique, Paris, 2013, 640 pages, € 40.00.

This unusual book is the outgrowth of a conference on the physics of sports held in Paris, at École Polytechnique, in April 2012. Observing that poetry, physics, and painting all have something to say about a stone and a tree, the brief introduction goes on to suggest that scientific analysis of unusual topics can be especially fruitful, as in the case of sports. As evidence, it cites a pair of 1970s papers on racing strategies by Joe Keller.

The book’s first chapter, titled “Physics ‘pour tout Le Monde,’” reproduces a dozen non-technical essays from the French daily Le Monde by journalists Pierre Lepidi and David Larrousserie. Serialized during the 2012 Olympics, the essays offer non-technical answers to a variety of FAQs, including: Why do race walkers flail their arms? Why do arrows warp? Why do sprinters have swollen calves? Each essay, reproduced both in the original French and in English translation, directs readers to a place in the book where the question is treated technically.

Each of the remaining six chapters consists of a brief introduction to a particular subdiscipline of physics, including aerodynamics, elasticity, and friction, followed by a series of studies in which the associated methods are applied to issues from the world of sports.

Underwater Swimming

Figure 1. Wave drag measurement apparatus: far from the surface (left), close to the surface (right).

In Chapter II, “Waves and Fluids,” the seventh section, titled “Wave Drag on the Swimmer,” relates to a Chapter I essay question: Why do we swim faster underwater? Experience shows that of the four Olympic strokes, at least two—backstroke and breaststroke—can be swum more rapidly underwater (for backstroke, using the dolphin kick, without arms) than on the surface. Accordingly, the Fédération Internationale de Natation (FINA) now imposes a rule permitting freestyle, breaststroke, and backstroke racers to remain totally submerged during the first 15 meters of each race, and again after each turn.

To quantify the advantages of swimming underwater, Clanet and two associates devised an experimental apparatus consisting of two smooth balls, mounted at opposite ends of a rod left free to pivot about its midpoint, as indicated in Figure 1. When the apparatus was towed at various speeds while deeply submerged, they observed little or no wave action on the surface and no discernible torque about the pivot. But at similar speeds nearer the surface, they observed unmistakable surface waves and significant torque about the pivot. Attributing the latter to “wave drag” on the upper ball, they noted that the torque increased to a maximum as the speed of the apparatus approached \(\sqrt{gD}\), where \(D\) is the diameter of the sphere and \(g\) the acceleration due to gravity, and decreased thereafter.

Thus, when wave drag force is plotted against the nondimensional Froude number \(Fr = \sqrt{(V/gD)}\), for various speeds, depths, and sphere diameters, the peaks all seem to cluster about the vertical line \(Fr = 1\). Perhaps more importantly, the torque is observed to vary inversely with the depth of submersion, all but vanishing when the upper ball lies more than a diameter below the surface.

Swimmers, of course, are not spherical. But additional experiments suggest that they too experience significant wave drag when swimming just below the surface, and that such drag all but vanishes when they are submerged by more than the thickness of a typical human body. Indeed, taking a swimmer’s shoulder-to-shoulder breadth as his or her characteristic length \(D\), the foregoing conclusions appear to apply nearly as well to ordinary swimmers as to spherical ones! Given that (due to fatigue) drag effects are more important in long races than in short ones, it is hardly surprising that long-distance swimmers (up to and including the incomparable Michael Phelps) seem more discerning than sprinters about the depths at which they swim.

Ballgame Dynamics

The chapter on aerodynamics includes several studies of balls in flight, both with and without spin. “The Aerodynamics of the Beautiful Game,” by J.W.M. Bush, examines the flight characteristics of soccer balls.¹ Elsewhere in the chapter are studies of ski jumping, shuttlecocks, tennis strokes, and discus throwing, along with a surprising one—for a book published in France—called “What New Technologies Are Teaching Us About the Game of Baseball,” by A.M. Nathan.

Figure 2. A fly ball trajectory (solid curve), with the initial conditions shown; the dots show the location of the ball at 0.5-sec intervals. Also shown are trajectories for the same initial conditions and neither drag nor Magnus force (short-dashed curve) and with drag but no Magnus (long-dashed curve).

The fruitful new technologies are of two kinds. In the so-called f/x systems, two 60-Hz digital cameras mounted high above the playing field send images to a nearby desktop computer. The cameras are mounted such that their principal axes meet roughly halfway between the pitcher’s mound and home plate. Proprietary software determines the camera coordinates of the baseball in each image and converts them into a location on or above the field. The most common of these systems, known as PITCHf/x, has been installed in every Major League Baseball park since 2007. The HITf/x system uses the same cameras to track the initial trajectories of batted balls, including their speeds, their vertical launch angles, and their “spray angles” to the right or left of the line through home plate and second base. The more ambitious FIELDf/x is intended to track almost everything that moves on the field, including the fielders, base runners, umpires, and batted or thrown balls; to date it has been installed at a few parks.

The rival TrackMan system is a phased-array Doppler radar installation that is also in use at a number of Major League parks. When installed high above home plate, its field of vision covers most of the playing field. For pitched balls, it is about as accurate as PITCHf/x; for batted balls its accuracy is as yet undetermined.

Because the drag and Magnus forces on baseballs in flight can be nearly as strong as the gravitational force, it seems unlikely that either can be ignored in any serious study of baseball trajectories. To dramatize the effects of those two forces, Nathan considers the flight of an actual home run that came off the bat with an initial speed of 112 mph, at a launch angle of 27° above the horizontal, with initial backspin of 1286 rpm. In the absence of both drag and Magnus forces, such a ball would travel nearly 700 feet. In the event, it travelled only 430 feet. Figure 2 suggests that, had it been subject to drag alone, it would have gone only about 380 feet. The Magnus force from backspin caused it to fly some 30 feet higher and 50 feet farther.

Figure 3. Typical sequence of actions in the pole vault. Reproduced in Sports Physics from Ganslen (1979).

Much has been learned about the flight of baseballs through the use of these and other technologies, and Nathan’s article rather neatly summarizes the current state of the art. Still open are questions regarding the onset of the so-called drag crisis, wherein the drag coefficient CD decreases temporarily as a function of airspeed. For speeds lower than 40 mph, the coefficient is about 0.50, while for speeds on the order of 90 mph, it falls to about 0.30. But experiments designed to determine the precise speeds at which the reduction occurs appear to disagree, leaving the phenomenon poorly understood.

Elasticity in the High Jump and Pole Vault

The chapter on elasticity includes analyses of high jumping and pole vaulting (“Energy Transformation in the Pole Vault”). Unlike the bamboo and aluminum poles of old, modern fiberglass poles are highly flexible. In bending, they acquire significant quantities of potential energy, which can be transferred to the jumper in the form of kinetic energy, as suggested by Figure 3.

Figure 4. When the jumper's center of mass CM reaches the zenith Z of its parabolic trajectory, the jumper's hinge-point H would lie 10 inches above Z and 1 inch above the bar.

Among the more unexpected revelations of the book’s discussion of high jumping and pole vaulting is that, whereas a competitor’s entire body must pass over the bar, his or her center of mass (CM) need not. To see why, imagine the human body to consist of a rod 80 inches long—high jumpers tend to be tall—and hinged in the middle. Once airborne, the rod’s CM cannot deviate from its initial parabolic path. So if the zenith of that path lies 9 inches below the bar, and if the rod is instantaneously bent at an angle of 120°, with one segment on either side of the bar, the hinge point H will stand 1 inch above the bar, as indicated in Figure 4.

Still to be explained is how a jumper should contort his or her body while airborne in order to reach such a position above the bar, but the drawing in Figure 3 looks enough like the stop-action photos taken of actual world-class high-jumpers to suggest that at least a few of them have mastered the trick. In practice, only a few of the most gifted seem to have done so.

All in all, Clanet has assembled an extraordinarily informative volume – one that belongs on the shelf of anyone seriously interested in understanding athletic endeavor.

¹ It can be seen as a precursor of the soccer ball study described in the June 2014 issue of SIAM News.

James Case writes from Baltimore, Maryland.